 5.1: For the nonlinear system dx/dt = x + x y and dy/dt = y2 2y, what is...
 5.2: For the nonlinear system in Exercise 1, determine the type (sink, s...
 5.3: For the nonlinear system dx/dt = x2 + sin 3x and dy/dt = 2y sin x y...
 5.4: For the nonlinear system in Exercise 3, determine the type (sink, s...
 5.5: Sketch the nullclines for the system dx/dt = x y and dy/dt = x2 + y...
 5.6: Is the system dx/dt = 2x y + y2 and dy/dt = x2 + y2 a Hamiltonian s...
 5.7: Is the system in Exercise 6 a gradient system?
 5.8: Describe three different possible longterm behaviors for a solutio...
 5.9: Suppose that a twodimensional system dY/dt = F(Y) has exactly one ...
 5.10: Suppose that the origin is an equilibrium point of a twodimensiona...
 5.11: The equilibrium points of a firstorder system occur at the interse...
 5.12: The x and ynullclines of a system are never identical.
 5.13: If (x0, y0) is an equilibrium point of the system dx/dt = f (x, y) ...
 5.14: For a system of the form dx/dt = f (x) and dy/dt = g(x, y) with f (...
 5.15: dx dt = x 3y2 dy dt = x 3y 6
 5.16: dx dt = 10 x2 y2 dy dt = 3x y
 5.17: dx dt = 4x x2 x y dy dt = 6y 2y2 x y
 5.18: dx dt = x y dy dt = x + y 1
 5.19: Find and classify all equilibrium points assuming that a > 0 and b ...
 5.20: Sketch the nullclines and the phase portrait assuming that a > 0 an...
 5.21: Repeat Exercises 19 and 20 assuming that a = 0 and b > 0. In additi...
 5.22: Repeat Exercises 19 and 20 assuming that a > 0 and b = 0. In additi...
 5.23: Repeat Exercises 19 and 20 assuming that a = 0 and b = 0.
 5.24: Draw a picture in the abplane that represents the different types ...
 5.25: Consider the system dx dt = y2 x2 1 dy dt = 2x y. (a) Find and clas...
 5.26: Consider the differential equation d2x dt2 + 2 dx dt 3x + x3 = 0. (...
 5.27: In Exercise 26 of Section 3.4, we showed that the solution curves o...
 5.28: Consider the linear system dY dt = a b c d Y. (a) For which values ...
 5.29: In Exercise 37 of the Review Exercises of Chapter 2, we studied a s...
Solutions for Chapter 5: Nonlinear Systems
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 5: Nonlinear Systems
Get Full SolutionsDifferential Equations 00 was written by and is associated to the ISBN: 9780495561989. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5: Nonlinear Systems includes 29 full stepbystep solutions. Since 29 problems in chapter 5: Nonlinear Systems have been answered, more than 16378 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.